Deterministic trend

*The author of this computation has been verified*

R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Fri, 19 Nov 2010 12:13:34 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Nov/19/t1290168856so3ozdxpfi23hvy.htm/, Retrieved Fri, 19 Nov 2010 13:14:22 +0100

BibTeX entries for LaTeX users:

@Manual{KEY, author = {{YOUR NAME}}, publisher = {Office for Research Development and Education}, title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2010/Nov/19/t1290168856so3ozdxpfi23hvy.htm/}, year = {2010}, } @Manual{R, title = {R: A Language and Environment for Statistical Computing}, author = {{R Development Core Team}}, organization = {R Foundation for Statistical Computing}, address = {Vienna, Austria}, year = {2010}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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6 101,82 107,34 93,63 99,85 101,76 6 101,68 107,34 93,63 99,91 102,37 6 101,68 107,34 93,63 99,87 102,38 6 102,45 107,34 96,13 99,86 102,86 6 102,45 107,34 96,13 100,10 102,87 6 102,45 107,34 96,13 100,10 102,92 6 102,45 107,34 96,13 100,12 102,95 6 102,45 107,34 96,13 99,95 103,02 6 102,45 112,60 96,13 99,94 104,08 6 102,52 112,60 96,13 100,18 104,16 6 102,52 112,60 96,13 100,31 104,24 6 102,85 112,60 96,13 100,65 104,33 7 102,85 112,61 96,13 100,65 104,73 7 102,85 112,61 96,13 100,69 104,86 7 103,25 112,61 96,13 101,26 105,03 7 103,25 112,61 98,73 101,26 105,62 7 103,25 112,61 98,73 101,38 105,63 7 103,25 112,61 98,73 101,38 105,63 7 104,45 112,61 98,73 101,38 105,94 7 104,45 112,61 98,73 101,44 106,61 7 104,45 118,65 98,73 101,40 107,69 7 104,80 118,65 98,73 101,40 107,78 7 104,80 118,65 98,73 100,58 107,93 7 105,29 118,65 98,73 100,58 108,48 8 105,29 114,29 98,73 100,58 108,14 8 105,29 114,29 98,73 100,59 108,48 8 105,29 114,29 98,73 100,81 108,48 8 106,04 114,29 101,67 100,75 1 etc...

Output produced by software:

Enter (or paste) a matrix (table) containing all data (time) series. Every column represents a different variable and must be delimited by a space or Tab. Every row represents a period in time (or category) and must be delimited by hard returns. The easiest way to enter data is to copy and paste a block of spreadsheet cells. Please, do not use commas or spaces to seperate groups of digits! Summary of computational transaction Raw Input view raw input (R code) Raw Output view raw output of R engine Computing time 7 seconds R Server 'RServer@AstonUniversity' @ vre.aston.ac.uk Multiple Linear Regression - Estimated Regression Equation And.dienstenrecr.&cultuur[t] = + 67.2314876034129 + 0.0808949042160557maand[t] + 0.111902922150727Bioscoop[t] + 0.177271331756085Schouwburgabonnement[t] + 0.12737147086898Eendagsattracties[t] -0.0868387869852838HuurvaneenDVD[t] + 0.169151767769642t + e[t] Multiple Linear Regression - Ordinary Least Squares Variable Parameter S.D. T-STAT H0: parameter = 0 2-tail p-value 1-tail p-value (Intercept) 67.2314876034129 8.061054 8.3403 0 0 maand 0.0808949042160557 0.15474 0.5228 0.603391 0.301695 Bioscoop 0.111902922150727 0.020322 5.5064 1e-06 1e-06 Schouwburgabonnement 0.177271331756085 0.021622 8.1987 0 0 Eendagsattracties 0.12737147086898 0.032484 3.9211 0.000264 0.000132 HuurvaneenDVD -0.0868387869852838 0.09024 -0.9623 0.340437 0.170218 t 0.169151767769642 0.020911 8.0889 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.998679902214923 R-squared 0.997361547088008 Adjusted R-squared 0.997051140863068 F-TEST (value) 3213.08487701928 F-TEST (DF numerator) 6 F-TEST (DF denominator) 51 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.302735427596801 Sum Squared Residuals 4.67408569523312 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 101.76 101.563207017546 0.196792982453787 2 102.37 101.711482048995 0.658517951004512 3 102.38 101.884107368245 0.495892631755449 4 102.86 102.458721451113 0.401278548887449 5 102.87 102.607031910006 0.26296808999428 6 102.92 102.776183677775 0.143816322224634 7 102.95 102.943598669805 0.00640133019470001 8 103.02 103.127513031362 -0.107513031362447 9 104.08 104.229980392039 -0.149980392038948 10 104.16 104.386124055483 -0.226124055482673 11 104.24 104.543986780944 -0.30398678094423 12 104.33 104.720541325449 -0.390541325448612 13 104.73 104.972360710752 -0.242360710751866 14 104.86 105.138038927042 -0.278038927042102 15 105.03 105.30245375509 -0.272453755090422 16 105.62 105.802771347119 -0.182771347119409 17 105.63 105.961502460451 -0.331502460450827 18 105.63 106.13065422822 -0.500654228220469 19 105.94 106.434089502571 -0.494089502570982 20 106.61 106.598030943122 0.0119690568784952 21 107.69 107.841375106177 -0.151375106177316 22 107.78 108.0496928967 -0.269692896699708 23 107.93 108.290052469797 -0.360052469797278 24 108.48 108.514036669421 -0.0340366694207803 25 108.14 107.99118033495 0.14881966505005 26 108.48 108.15946371485 0.320536285150265 27 108.48 108.309510949483 0.170489050517385 28 108.89 108.942272360439 -0.0522723604392227 29 108.93 109.100233835994 -0.170233835993785 30 109.21 109.251149458497 -0.0411494584965309 31 109.47 109.389907650821 0.0800923491786822 32 109.8 109.568449612417 0.231550387582934 33 111.73 111.378145785349 0.351854214650791 34 111.85 111.60330215536 0.246697844640151 35 112.12 111.878761699173 0.241238300827328 36 112.15 112.064681191543 0.0853188084569534 37 112.17 112.303438821221 -0.133438821220662 38 112.67 112.518837892239 0.151162107761354 39 112.8 112.67496384196 0.1250361580395 40 113.44 113.642734732079 -0.202734732078643 41 113.53 113.808412948369 -0.27841294836887 42 114.53 114.027921031106 0.502078968893661 43 114.51 114.163205671952 0.346794328048282 44 115.05 114.757606257616 0.292393742383994 45 116.67 116.161240739081 0.508759260918836 46 117.07 116.541131279776 0.5288687202239 47 116.92 116.712019823285 0.20798017671456 48 117 116.90928149913 0.0907185008704701 49 117.02 117.172756521773 -0.15275652177332 50 117.35 117.373491749097 -0.0234917490968216 51 117.36 117.662015187647 -0.302015187646592 52 117.82 118.131664876574 -0.311664876574018 53 117.88 118.314612100168 -0.434612100168289 54 118.24 118.541335640938 -0.301335640938112 55 118.5 118.713593855021 -0.213593855020616 56 118.8 118.891429501489 -0.0914295014887889 57 119.76 119.746621323154 0.0133786768455259 58 120.09 119.907089212226 0.18291078777441 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 10 0.112420926588102 0.224841853176203 0.887579073411898 11 0.0476101742046358 0.0952203484092716 0.952389825795364 12 0.226824863996481 0.453649727992962 0.773175136003519 13 0.137016396197795 0.274032792395591 0.862983603802205 14 0.0768929576814084 0.153785915362817 0.923107042318592 15 0.056912481426711 0.113824962853422 0.943087518573289 16 0.0412454120766411 0.0824908241532823 0.958754587923359 17 0.0255529792967069 0.0511059585934138 0.974447020703293 18 0.0230581027798156 0.0461162055596312 0.976941897220184 19 0.0286498153864868 0.0572996307729737 0.971350184613513 20 0.22815237999668 0.45630475999336 0.77184762000332 21 0.274059147912992 0.548118295825985 0.725940852087008 22 0.271369270516563 0.542738541033127 0.728630729483437 23 0.312026723424919 0.624053446849837 0.687973276575081 24 0.355381698770114 0.710763397540228 0.644618301229886 25 0.32063747362124 0.64127494724248 0.67936252637876 26 0.479181607158921 0.958363214317841 0.520818392841079 27 0.497548358899305 0.99509671779861 0.502451641100695 28 0.490891716538702 0.981783433077404 0.509108283461298 29 0.425269774691414 0.850539549382828 0.574730225308586 30 0.366770523694443 0.733541047388885 0.633229476305557 31 0.387496712763041 0.774993425526082 0.612503287236959 32 0.425619467204281 0.851238934408562 0.574380532795719 33 0.661252217570701 0.677495564858599 0.338747782429299 34 0.592120657924981 0.815758684150038 0.407879342075019 35 0.524422772577521 0.951154454844958 0.475577227422479 36 0.580118632948776 0.839762734102449 0.419881367051224 37 0.655190680535214 0.689618638929572 0.344809319464786 38 0.582566476572724 0.834867046854551 0.417433523427276 39 0.521416848573943 0.957166302852114 0.478583151426057 40 0.502458639916711 0.995082720166578 0.497541360083289 41 0.901180212205302 0.197639575589396 0.0988197877946978 42 0.905651051585639 0.188697896828723 0.0943489484143614 43 0.917404012542562 0.165191974914877 0.0825959874574383 44 0.885062565211635 0.22987486957673 0.114937434788365 45 0.824122136902134 0.351755726195731 0.175877863097866 46 0.983833829500144 0.0323323409997117 0.0161661704998558 47 0.983414399178296 0.0331712016434076 0.0165856008217038 48 0.944905285659128 0.110189428681744 0.0550947143408718 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 3 0.0769230769230769 NOK 10% type I error level 7 0.179487179487179 NOK

Charts produced by software:

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Parameters (Session):

par1 = 4 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 6 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

R code (references can be found in the software module):

library(lattice) library(lmtest) n25 <- 25 #minimum number of obs. for Goldfeld-Quandt test par1 <- as.numeric(par1) x <- t(y) k <- length(x[1,]) n <- length(x[,1]) x1 <- cbind(x[,par1], x[,1:k!=par1]) mycolnames <- c(colnames(x)[par1], colnames(x)[1:k!=par1]) colnames(x1) <- mycolnames #colnames(x)[par1] x <- x1 if (par3 == 'First Differences'){ x2 <- array(0, dim=c(n-1,k), dimnames=list(1:(n-1), paste('(1-B)',colnames(x),sep=''))) for (i in 1:n-1) { for (j in 1:k) { x2[i,j] <- x[i+1,j] - x[i,j] } } x <- x2 } if (par2 == 'Include Monthly Dummies'){ x2 <- array(0, dim=c(n,11), dimnames=list(1:n, paste('M', seq(1:11), sep =''))) for (i in 1:11){ x2[seq(i,n,12),i] <- 1 } x <- cbind(x, x2) } if (par2 == 'Include Quarterly Dummies'){ x2 <- array(0, dim=c(n,3), dimnames=list(1:n, paste('Q', seq(1:3), sep =''))) for (i in 1:3){ x2[seq(i,n,4),i] <- 1 } x <- cbind(x, x2) } k <- length(x[1,]) if (par3 == 'Linear Trend'){ x <- cbind(x, c(1:n)) colnames(x)[k+1] <- 't' } x k <- length(x[1,]) df <- as.data.frame(x) (mylm <- lm(df)) (mysum <- summary(mylm)) if (n > n25) { kp3 <- k + 3 nmkm3 <- n - k - 3 gqarr <- array(NA, dim=c(nmkm3-kp3+1,3)) numgqtests <- 0 numsignificant1 <- 0 numsignificant5 <- 0 numsignificant10 <- 0 for (mypoint in kp3:nmkm3) { j <- 0 numgqtests <- numgqtests + 1 for (myalt in c('greater', 'two.sided', 'less')) { j <- j + 1 gqarr[mypoint-kp3+1,j] <- gqtest(mylm, point=mypoint, alternative=myalt)$p.value } if (gqarr[mypoint-kp3+1,2] < 0.01) numsignificant1 <- numsignificant1 + 1 if (gqarr[mypoint-kp3+1,2] < 0.05) numsignificant5 <- numsignificant5 + 1 if (gqarr[mypoint-kp3+1,2] < 0.10) numsignificant10 <- numsignificant10 + 1 } gqarr } bitmap(file='test0.png') plot(x[,1], type='l', main='Actuals and Interpolation', ylab='value of Actuals and Interpolation (dots)', xlab='time or index') points(x[,1]-mysum$resid) grid() dev.off() bitmap(file='test1.png') plot(mysum$resid, type='b', pch=19, main='Residuals', ylab='value of Residuals', xlab='time or index') grid() dev.off() bitmap(file='test2.png') hist(mysum$resid, main='Residual Histogram', xlab='values of Residuals') grid() dev.off() bitmap(file='test3.png') densityplot(~mysum$resid,col='black',main='Residual Density Plot', xlab='values of Residuals') dev.off() bitmap(file='test4.png') qqnorm(mysum$resid, main='Residual Normal Q-Q Plot') qqline(mysum$resid) grid() dev.off() (myerror <- as.ts(mysum$resid)) bitmap(file='test5.png') dum <- cbind(lag(myerror,k=1),myerror) dum dum1 <- dum[2:length(myerror),] dum1 z <- as.data.frame(dum1) z plot(z,main=paste('Residual Lag plot, lowess, and regression line'), ylab='values of Residuals', xlab='lagged values of Residuals') lines(lowess(z)) abline(lm(z)) grid() dev.off() bitmap(file='test6.png') acf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Autocorrelation Function') grid() dev.off() bitmap(file='test7.png') pacf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Partial Autocorrelation Function') grid() dev.off() bitmap(file='test8.png') opar <- par(mfrow = c(2,2), oma = c(0, 0, 1.1, 0)) plot(mylm, las = 1, sub='Residual Diagnostics') par(opar) dev.off() if (n > n25) { bitmap(file='test9.png') plot(kp3:nmkm3,gqarr[,2], main='Goldfeld-Quandt test',ylab='2-sided p-value',xlab='breakpoint') grid() dev.off() } load(file='createtable') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Estimated Regression Equation', 1, TRUE) a<-table.row.end(a) myeq <- colnames(x)[1] myeq <- paste(myeq, '[t] = ', sep='') for (i in 1:k){ if (mysum$coefficients[i,1] > 0) myeq <- paste(myeq, '+', '') myeq <- paste(myeq, mysum$coefficients[i,1], sep=' ') if (rownames(mysum$coefficients)[i] != '(Intercept)') { myeq <- paste(myeq, rownames(mysum$coefficients)[i], sep='') if (rownames(mysum$coefficients)[i] != 't') myeq <- paste(myeq, '[t]', sep='') } } myeq <- paste(myeq, ' + e[t]') a<-table.row.start(a) a<-table.element(a, myeq) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable1.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,hyperlink('http://www.xycoon.com/ols1.htm','Multiple Linear Regression - Ordinary Least Squares',''), 6, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Variable',header=TRUE) a<-table.element(a,'Parameter',header=TRUE) a<-table.element(a,'S.D.',header=TRUE) a<-table.element(a,'T-STAT<br />H0: parameter = 0',header=TRUE) a<-table.element(a,'2-tail p-value',header=TRUE) a<-table.element(a,'1-tail p-value',header=TRUE) a<-table.row.end(a) for (i in 1:k){ a<-table.row.start(a) a<-table.element(a,rownames(mysum$coefficients)[i],header=TRUE) a<-table.element(a,mysum$coefficients[i,1]) a<-table.element(a, round(mysum$coefficients[i,2],6)) a<-table.element(a, round(mysum$coefficients[i,3],4)) a<-table.element(a, round(mysum$coefficients[i,4],6)) a<-table.element(a, round(mysum$coefficients[i,4]/2,6)) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable2.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Regression Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple R',1,TRUE) a<-table.element(a, sqrt(mysum$r.squared)) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'R-squared',1,TRUE) a<-table.element(a, mysum$r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Adjusted R-squared',1,TRUE) a<-table.element(a, mysum$adj.r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (value)',1,TRUE) a<-table.element(a, mysum$fstatistic[1]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF numerator)',1,TRUE) a<-table.element(a, mysum$fstatistic[2]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF denominator)',1,TRUE) a<-table.element(a, mysum$fstatistic[3]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'p-value',1,TRUE) a<-table.element(a, 1-pf(mysum$fstatistic[1],mysum$fstatistic[2],mysum$fstatistic[3])) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Residual Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Residual Standard Deviation',1,TRUE) a<-table.element(a, mysum$sigma) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Sum Squared Residuals',1,TRUE) a<-table.element(a, sum(myerror*myerror)) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable3.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Actuals, Interpolation, and Residuals', 4, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Time or Index', 1, TRUE) a<-table.element(a, 'Actuals', 1, TRUE) a<-table.element(a, 'Interpolation<br />Forecast', 1, TRUE) a<-table.element(a, 'Residuals<br />Prediction Error', 1, TRUE) a<-table.row.end(a) for (i in 1:n) { a<-table.row.start(a) a<-table.element(a,i, 1, TRUE) a<-table.element(a,x[i]) a<-table.element(a,x[i]-mysum$resid[i]) a<-table.element(a,mysum$resid[i]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable4.tab') if (n > n25) { a<-table.start() a<-table.row.start(a) a<-table.element(a,'Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'p-values',header=TRUE) a<-table.element(a,'Alternative Hypothesis',3,header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'breakpoint index',header=TRUE) a<-table.element(a,'greater',header=TRUE) a<-table.element(a,'2-sided',header=TRUE) a<-table.element(a,'less',header=TRUE) a<-table.row.end(a) for (mypoint in kp3:nmkm3) { a<-table.row.start(a) a<-table.element(a,mypoint,header=TRUE) a<-table.element(a,gqarr[mypoint-kp3+1,1]) a<-table.element(a,gqarr[mypoint-kp3+1,2]) a<-table.element(a,gqarr[mypoint-kp3+1,3]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable5.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,'Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Description',header=TRUE) a<-table.element(a,'# significant tests',header=TRUE) a<-table.element(a,'% significant tests',header=TRUE) a<-table.element(a,'OK/NOK',header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'1% type I error level',header=TRUE) a<-table.element(a,numsignificant1) a<-table.element(a,numsignificant1/numgqtests) if (numsignificant1/numgqtests < 0.01) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'5% type I error level',header=TRUE) a<-table.element(a,numsignificant5) a<-table.element(a,numsignificant5/numgqtests) if (numsignificant5/numgqtests < 0.05) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'10% type I error level',header=TRUE) a<-table.element(a,numsignificant10) a<-table.element(a,numsignificant10/numgqtests) if (numsignificant10/numgqtests < 0.1) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable6.tab') }

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